Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

Abstract

The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

Modeling two loops RLC circuit AC power source using symbolic arithmetic differential equations

“…Deep learning-based data assimilation algorithms [21], [22] were lately presented to train Navier-Stokes network formulas to estimate different quantities of interesting. DLCNN-based methods can be classified into three kinds: i) DLCNN that maps indirectly to the parameters or inner results of an algebraic explanation to use them for deriving numerical solutions [23], [24]; ii) DLCNN that maps directly toward the solutions with a discrete method, which is same as numerical solutions [25], [26]; and iii) DLCNN that maps straight to the solution characterized by a DLCNN with a continuous way and it is similar to that in analytical solutions [27]- [29]. In this type, the data applied for training the network are arbitrarily modeled inside the whole solution range in every training batch, including boundary and initial circumstances.…”